Shear Force and Bending Moment Diagrams

Introduction

• Essential tools for mechanical and civil engineers to analyze beams under loading
• Aim: Master shear force and bending moment diagrams to understand how beams are loaded

Understanding Shear Forces and Bending Moments

• Internal Forces in Beams
  • Maintain equilibrium when a beam is loaded
  • Two components: Shear Forces and Normal Forces
• Shear Forces
  • Oriented vertically
• Normal Forces
  • Oriented along the beam's axis
  • Top of a sagging beam: Compressive forces
  • Bottom of a sagging beam: Tensile forces
  • Tensile normal forces have corresponding compressive forces (equal in magnitude, opposite in direction)
• Resultants
  • Shear Force: Resultant of vertical internal forces
  • Bending Moment: Resultant of normal internal forces

Representing Internal Forces

• Representing using just two resultants: Shear force and bending moment
• Figures out internal forces at each location along the beam
• Dependent on loads and supports

Types of Loads and Supports

• Common Loads
  • Concentrated forces
  • Distributed forces
  • Concentrated moments
• Common Supports
  • Pinned Support: Prevents vertical/horizontal displacements, allows rotation
  • Roller Support: Prevents vertical displacement, allows horizontal displacement and rotation
  • Fixed Support: Prevents all displacements and rotation
• Reaction Forces and Moments
  • Reaction forces/moment at points of restraint
  • Example: Pinned support permits rotation (no reaction moment), restrains vertical/horizontal displacements (reaction forces are present)

Determining Shear Forces and Bending Moments

• Three Main Steps
  1. Draw the free body diagram (FBD) of the beam
  2. Calculate reaction forces and reaction moments using equilibrium equations
  3. Compute internal shear forces and bending moments along the beam

Equilibrium Equations

• Maintain equilibrium via vertical/horizontal forces and moments
• Statically determinate: Can calculate all reaction loads with equilibrium equations
• Statically indeterminate: More unknowns than equilibrium equations, need advanced methods and boundary conditions
• Focus: Statically determinate cases in this lecture

Calculating Internal Forces and Moments

• Using Equilibrium
  • Cut the beam at a location
  • Ensure internal forces/moments cancel out external forces/moments for equilibrium

Sign Convention

• Applied Forces: Positive if downward
• Shear Forces: Positive if downward on the left side or upward on the right side of the cut
• Bending Moments: Positive if they cause sagging (lower section in tension)

Example Calculation

• Free Body Diagram of a Pinned and Roller Supported Beam
  • Two concentrated loads
• Equilibrium Equations
  • Sum of vertical forces = 0: R-A + R-B = 15 + 6
  • Horizontal force (H-A) = 0 (only horizontal force)
  • Sum of moments about point (e.g., Point B) = 0
  • Determine R-A and then R-B
• Drawing Shear Force and Bending Moment Diagrams
  • Calculate shear forces to the right of each applied force and draw the FBD
  • Repeat process moving the cut along the beam

Relationship Between Loads, Shear Forces, and Bending Moments

• Consider arbitrary distributed force
• Infinitesimally Small Segment Analysis
  • Distributed force -> Equivalent concentrated force
  • Equilibrium for small segment: Relate distributed force, shear force, and bending moment
• Resulting Equations
  • Slope of shear force curve = -distributed force
  • Slope of bending moment curve = shear force
  • Change in shear force = Area under loading diagram
  • Change in bending moment = Area under shear force curve

Advanced Example

• Combined Loads: Distributed and concentrated
• Deriving Bending Moment Equation: Bending moment under distributed force gives quadratic equation
• Differentiation
  • First derivative: Equation for shear force curve
  • Second derivative: Equation for distributed force
• Area Calculation Method
  • Area under shear force curve = Change in bending moment
  • Validate diagrams via area calculations
• Concentrated Forces / Moments
  • Sudden jumps in shear force / bending moment diagrams

Cantilever Beam Example

• Setup: Concentrated moment and distributed force
• Free Body Diagram
  • Fixed support with reaction forces and moment
• Calculations
  • Vertical/horizontal forces and moments for equilibrium
  • Compute shear force and bending moment moving along beam
  • Derive equations for different segments of the beam

Deformed Shape Prediction

• Use bending moment information
  • Positive bending moment: Sagging
  • Negative bending moment: Hogging
  • Zero bending moment: Straight sections

Conclusion

• Review: Internal forces, equilibrium, drawing diagrams, advanced checks
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